1. 1.7 – Linear Inequalities and Compound Inequalities
Properties of Inequalities
Addition and Subtraction Property of Inequality
If a < b, then a + c < b + c or If a > b, then a + c > b + c
If a < b, then a - c < b - c or If a > b, then a - c > b - c
Multiplication and Division Property of Inequality
c is positive:
If a < b, then a • c < b • c or If a > b, then a • c > b • c
If a < b, then a/c < b/c or If a > b, then a/c > b/c
c is negative:
If a < b, then a • c > b • c or If a > b, then a • c < b • c
If a < b, then a/c > b/c or If a > b, then a/c < b/c

2. 1.7 – Linear Inequalities and Compound Inequalities
Solving Inequalities
Examples:
4𝑥−9+3𝑥≤2𝑥−5+7𝑥
−7 𝑥+9 ≥40+3𝑥
7𝑥−9≤9𝑥−5
−7𝑥−63≥40+3𝑥
−2𝑥≤4
−10𝑥≥103
𝑥≥−2
𝑥≤−10.3 -3 -2 -1 9 10 11
−2, ∞
−∞, −10.3

3. 1.7 – Linear Inequalities and Compound Inequalities
Solving Inequalities
Examples:
−2 2−2𝑥 −4 𝑥+5 ≤−24
3 1−2𝑥 >8−6𝑥
−4+4𝑥−4𝑥−20≤−24
3−6𝑥>8−6𝑥
−24≤−24
3>8
(1) Lost the variable
(1) Lost the variable
(2) True statement
(2) False statement
∴ Solution: All Reals
∴ Solution: the null set -1 1 -1 1
−∞, ∞
∅ or { }

4. Properties of Inequalities Union and Intersection of Sets
1.7 – Linear Inequalities and Compound Inequalities
Properties of Inequalities
Union and Intersection of Sets
The Union of sets A and B 𝐴∪𝐵 represents the elements that are in either set.
The Intersection of sets A and B 𝐴∩𝐵 represents the elements that are common to both sets.
Examples: Determine the solution for each set operation.
𝐴: 𝑥|𝑥≥5
𝐵: 𝑥|3≤𝑥<12
𝐶: 𝑥|𝑥<−1 ) ) 5 3 12 -1
𝐴∩𝐵
𝐴∪𝐵
𝐵∩𝐶
𝐴∪𝐶
5, 12
3, ∞ ∅
−∞, −1 ∪ 5, ∞

5. Compound Inequalities
1.7 – Linear Inequalities and Compound Inequalities
Compound Inequalities
Example:
7𝑣−5≥ 𝑜𝑟 −3𝑣−2≥−2
7𝑣−5≥65
−3𝑣−2≥−2
7𝑣≥70
−3𝑣≥0
𝑣≥10
𝑣≤0
−∞, 0 ∪ 10, ∞

6. Compound Inequalities
1.7 – Linear Inequalities and Compound Inequalities
Compound Inequalities
Example:
8𝑥+8≥− 𝑎𝑛𝑑 −7−8𝑥≥−79
8𝑥+8≥−64
−7−8𝑥≥−79
8𝑥≥−72
−8𝑥≥−72
𝑥≥−9
𝑥≤9
−9, 9

7. Absolute Value Equations Properties of Absolute Values Equations
1.8 – Absolute Value Equations and Inequalities
Absolute Value Equations
Properties of Absolute Values Equations
𝒖 =𝒂
𝑎<0
No Solution
𝑎=0
One solution: 𝑢=0
𝑎>0
Two Solutions: 𝑢=𝑎 𝑜𝑟 𝑢=−𝑎
𝑢 = 𝑚
𝑢=𝑤 or u=−w
𝑢 = 𝑤
±𝑢=±𝑤
+𝑢=+𝑤
+𝑢=−𝑤
−𝑢=+𝑤
−𝑢=−𝑤
𝑢=𝑤
𝑢=−𝑤
𝑢=−𝑤
𝑢=𝑤
𝑢=𝑤
𝑢=−𝑤

8. Absolute Value Equations
1.8 – Absolute Value Equations and Inequalities
Absolute Value Equations
Examples:
−2𝑛+6 =6
−2𝑛+6=6
−2𝑛+6=−6
−2𝑛=0
−2𝑛=−12
𝑛=0
𝑛=6
𝑛=0, 6
𝑥+8 −5=2
𝑥+8 =7
𝑥+8=−7
𝑥+8=7
𝑥=−15
𝑥=−1
𝑥=−15, −1

9. Absolute Value Equations
1.8 – Absolute Value Equations and Inequalities
Absolute Value Equations
Example:
3 3−5𝑟 −3=18
3 3−5𝑟 =21
3−5𝑟 =7
3−5𝑟=−7
3−5𝑟=7
−5𝑟=−10
−5𝑟=4
𝑟=2
𝑟=− 4 5
𝑟=− 4 5 , 2

10. Absolute Value Equations
1.8 – Absolute Value Equations and Inequalities
Absolute Value Equations
Example:
5 9−5𝑛 −7=38
5 9−5𝑛 =45
9−5𝑛 =9
9−5𝑛=−9
9−5𝑛=9
−5𝑛=−18
−5𝑛=0
𝑛= 18 5
𝑛=0
𝑛=2, 18 5

11. Absolute Value Equations
1.8 – Absolute Value Equations and Inequalities
Absolute Value Equations
Example:
2𝑥−1 = 4𝑥+9
2𝑥−1=4𝑥+9
2𝑥−1=− 4𝑥+9
−2𝑥=10
2𝑥−1=−4𝑥−9
𝑥=−5
6𝑥=−8
𝑥=− 8 6 =− 4 3
𝑥=−5, − 4 3

12. Absolute Value Inequalities Properties of Absolute Values Inequalities
1.8 – Absolute Value Equations and Inequalities
Absolute Value Inequalities
Properties of Absolute Values Inequalities
𝑢 <𝑎
−𝑎<𝑢<𝑎
𝑢 ≤𝑎
−𝑎≤𝑢≤𝑎
𝑢 >𝑎
𝑢<−𝑎 𝑜𝑟 𝑢>𝑎
𝑢 ≥𝑎
𝑢≤−𝑎 𝑜𝑟 𝑢≥𝑎
10𝑦−4 <34
10𝑦−4>−34
10𝑦−4<34
10𝑦>−30
10𝑦<38
𝑦>−3
𝑦< 𝑜𝑟 𝑦<3.8
−3<𝑦<3.8
−3, 3.8

13. Absolute Value Inequalities Properties of Absolute Values Inequalities
1.8 – Absolute Value Equations and Inequalities
Absolute Value Inequalities
Properties of Absolute Values Inequalities
𝑢 <𝑎
−𝑎<𝑢<𝑎
𝑢 ≤𝑎
−𝑎≤𝑢≤𝑎
𝑢 >𝑎
𝑢<−𝑎 𝑜𝑟 𝑢>𝑎
𝑢 ≥𝑎
𝑢≤−𝑎 𝑜𝑟 𝑢≥𝑎
−8𝑥−3 >11
−8𝑥−3<−11
−8𝑥−3>11
−8𝑥<−8
−8𝑥>14
𝑥<− 𝑜𝑟 𝑥>1
𝑥>1
𝑥<− 14 8
−∞, − 7 4 ∪ 1, ∞
𝑥<− 7 4

14. Absolute Value Inequalities Properties of Absolute Values Inequalities
1.8 – Absolute Value Equations and Inequalities
Absolute Value Inequalities
Properties of Absolute Values Inequalities
𝑢 <𝑎
−𝑎<𝑢<𝑎
𝑢 ≤𝑎
−𝑎≤𝑢≤𝑎
𝑢 >𝑎
𝑢<−𝑎 𝑜𝑟 𝑢>𝑎
𝑢 ≥𝑎
𝑢≤−𝑎 𝑜𝑟 𝑢≥𝑎
4 6−2𝑎 +8≤24
4 6−2𝑎 ≤16
6−2𝑎 ≤4
6−2𝑎≥−4
6−2𝑎≤4
−2𝑎≥−10
−2𝑎≤−2
𝑎≤5
𝑎≥1
1≤𝑎≤5
[1, 5]

15. Absolute Value Inequalities Properties of Absolute Values Inequalities
1.8 – Absolute Value Equations and Inequalities
Absolute Value Inequalities
Properties of Absolute Values Inequalities
𝑢 <𝑎
−𝑎<𝑢<𝑎
𝑢 ≤𝑎
−𝑎≤𝑢≤𝑎
𝑢 >𝑎
𝑢<−𝑎 𝑜𝑟 𝑢>𝑎
𝑢 ≥𝑎
𝑢≤−𝑎 𝑜𝑟 𝑢≥𝑎
9 𝑟−2 −10<−73
9 𝑟−2 <−63
𝑟−2 <−7
𝑎𝑏𝑠𝑜𝑙𝑢𝑡𝑒 𝑣𝑎𝑙𝑢𝑒 𝑐𝑎𝑛𝑛𝑜𝑡 𝑏𝑒 𝑛𝑒𝑔𝑎𝑡𝑖𝑣𝑒
∴𝑛𝑜 𝑠𝑜𝑙𝑢𝑡𝑖𝑜𝑛

16. Absolute Value Inequalities Properties of Absolute Values Inequalities
1.8 – Absolute Value Equations and Inequalities
Absolute Value Inequalities
Properties of Absolute Values Inequalities
𝑢 <𝑎
−𝑎<𝑢<𝑎
𝑢 ≤𝑎
−𝑎≤𝑢≤𝑎
𝑢 >𝑎
𝑢<−𝑎 𝑜𝑟 𝑢>𝑎
𝑢 ≥𝑎
𝑢≤−𝑎 𝑜𝑟 𝑢≥𝑎
5 3𝑛−2 +6≥51
5 3𝑛−2 ≥45
3𝑛−2 ≥9
3𝑛−2≤−9
3𝑛−2≥9
3𝑛≤−7
3𝑛≥11
𝑛≤− 7 3
𝑛≥ 11 3
−∞, − 7 3 ∪ , ∞
𝑛≤− 𝑜𝑟 𝑛≥ 11 3